Page 79 - Krugmans Economics for AP Text Book_Neat

When variables are related in this way—that is, when an increase in one variable is asso-
ciated with an increase in the other variable—the variables are said to have a positive rela-
tionship. It is illustrated by a curve that slopes upward from left to right. Because this
curve is also linear, the relationship between outside temperature and number of sodas
sold illustrated by the curve in panel (a) of Figure A.2 is a positive linear relationship.
When an increase in one variable is associated with a decrease in the other variable,
the two variables are said to have a negative relationship. It is illustrated by a curve Section I Basic Economic Concepts
that slopes downward from left to right, like the curve in panel (b) of Figure A.2. Be-
cause this curve is also linear, the relationship it depicts is a negative linear relation-
ship. Two variables that might have such a relationship are the outside temperature
and the number of hot drinks a vendor can expect to sell at a baseball stadium.
Return for a moment to the curve in panel (a) of Figure A.2, and you can see that it
hits the horizontal axis at point B. This point, known as the horizontal intercept,
shows the value of the x-variable when the value of the y-variable is zero. In panel (b) of
Figure A.2, the curve hits the vertical axis at point J. This point, called the vertical in-
tercept, indicates the value of the y-variable when the value of the x-variable is zero.

A Key Concept: The Slope of a Curve

The slope of a curve is a measure of how steep it is; the slope indicates how sensitive the
y-variable is to a change in the x-variable. In our example of outside temperature and
the number of cans of soda a vendor can expect to sell, the slope of the curve would indi-
cate how many more cans of soda the vendor could expect to sell with each 1° increase in
temperature. Interpreted this way, the slope gives meaningful information. Even without
numbers for x and y, it is possible to arrive at important conclusions about the relation-
ship between the two variables by examining the slope of a curve at various points.

The Slope of a Linear Curve
Along a linear curve the slope, or steepness, is measured by dividing the “rise” between
two points on the curve by the “run” between those same two points. The rise is the
amount that y changes, and the run is the amount that x changes. Here is the formula:

Change in y = Δy = Slope
Change in x Δx

In the formula, the symbol Δ (the Greek uppercase delta) stands for “change in.”
When a variable increases, the change in that variable is positive; when a variable de-
creases, the change in that variable is negative.
The slope of a curve is positive when the rise (the change in the y-variable) has the
same sign as the run (the change in the x-variable). That’s because when two numbers
have the same sign, the ratio of those two numbers is positive. The curve in panel (a) of
Figure A.2 has a positive slope: along the curve, both the y-variable and the x-variable
increase. The slope of a curve is negative when the rise and the run have different signs.
That’s because when two numbers have different signs, the ratio of those two numbers
is negative. The curve in panel (b) of Figure A.2 has a negative slope: along the curve, an
increase in the x-variable is associated with a decrease in the y-variable.
Figure A.3 illustrates how to calculate the slope of a linear curve. Let’s focus first on
panel (a). From point A to point B the value of the y-variable changes from 25 to 20 and
the value of the x-variable changes from 10 to 20. So the slope of the line between these
two points is

Change in y = Δy = −5 = − = −0.5
1
Change in x Δx 10 2
Because a straight line is equally steep at all points, the slope of a straight line is the
same at all points. In other words, a straight line has a constant slope. You can check

appendix Graphs in Economics 37